# Power, Indices & Surds – Solved Questions

Question 01
Represent the following in form of perfect square
(4+ \sqrt{7}) \\\ \\ (a) \ (2+ \sqrt{7}) ^{2} \\\ \\ (b) \{ \frac{\sqrt{7}}{2} +\frac{1}{2}\} ^{2} \\\ \\ (c) \left\{\frac{1}{\sqrt{2}}\left(\sqrt{7} +1\right)\right\}^{2} \\\ \\ (d) ( \sqrt{3} +\sqrt{4}) ^{2}

4+ \sqrt{7} \\\ \\ ⟹ \frac{8+2\sqrt{7}}{2} \\\ \\ ⟹ \frac{\left(\sqrt{7}\right)^{2} +1+2\times \sqrt{7} \times 1}{2} \\\ \\ ⟹ \frac{\left(\sqrt{7} +1\right)^{2}}{\left(\sqrt{2}\right)^{2}}=\displaystyle \left[\frac{1}{\sqrt{2}}\left(\sqrt{7} +1\right)\right]^{2} \\\ \\

Option (b) is the right answer

Question 02
The simplified form of
\frac{2}{\sqrt{7} +\sqrt{5}} +\frac{7}{\sqrt{12} -\sqrt{5}} -\frac{5}{\sqrt{12} -\sqrt{7}} \ is \\\ \\ (a) 5 \\ \\ (b) 2 \\ \\ (c) 1 \\ \\ (d) 0

\frac{2}{\sqrt{7} +\sqrt{5}} +\frac{7}{\sqrt{12} -\sqrt{5}} -\frac{5}{\sqrt{12} -\sqrt{7}} \\\ \\ ⟹ \frac{2}{\sqrt{7} +\sqrt{5}} \times \frac{\sqrt{7} -\sqrt{5}}{\sqrt{7} -\sqrt{5}} +\frac{7}{\sqrt{12} -\sqrt{5}} \times \frac{\sqrt{12} +\sqrt{5}}{\sqrt{12} +\sqrt{5}} -\frac{5}{\sqrt{12} -\sqrt{7}} \times \frac{\sqrt{12} +\sqrt{7}}{\sqrt{12} +\sqrt{7}} \\\ \\\ ⟹ \frac{2\left(\sqrt{7} -\sqrt{5}\right)}{7+\sqrt{35} -\sqrt{35} -5} +\frac{7\left(\sqrt{12} +\sqrt{5}\right)}{12+\sqrt{60} -\sqrt{60} -5} -\frac{5\left(\sqrt{12} +\sqrt{7}\right)}{12\ +\sqrt{84} -\sqrt{84} +7} \\\ \\ ⟹ \frac{2\left(\sqrt{7} -\sqrt{5}\right)}{2} +\frac{7\left(\sqrt{12} +\sqrt{5}\right)}{7} -\frac{5\left(\sqrt{12} +\sqrt{7}\right)}{5} \\\ \\ ⟹ \sqrt{7} -\sqrt{5} +\sqrt{12} +\sqrt{5} -\sqrt{12} -\sqrt{7} \\\ \\ ⟹ 0

Option (d) is the right answer

Question 03
\frac{1}{\sqrt{3} +\sqrt{4}} +\frac{1}{\sqrt{4} +\sqrt{5}} +\frac{1}{\sqrt{5} +\sqrt{6}} +\frac{1}{\sqrt{6} +\sqrt{7}} +\frac{1}{\sqrt{7} +\sqrt{8}} +\frac{1}{\sqrt{8} +\sqrt{9}} is \\\ \\ (a) \sqrt{3} \\ \\ (b) 3 \sqrt{3} \\ \\ (c) 3- \sqrt{3} \\ \\ (d) 5- \sqrt{3}

\frac{1}{\sqrt{3} +\sqrt{4}} \\\ \\ ⟹ \frac{1}{\sqrt{4} +\sqrt{3}} \times \frac{\sqrt{4} -\sqrt{3}}{\sqrt{4} -\sqrt{3}} \\\ \\ ⟹ \frac{\sqrt{4} -\sqrt{3}}{4-\sqrt{12} +\sqrt{12} -3} \\\ \\ ⟹ \sqrt{4} -\sqrt{3} \\\ \\ \\\ \\ Similarly, \frac{1}{\sqrt{5} +\sqrt{4}}= \sqrt{5} -\sqrt{4} \\\ \\ \frac{1}{\sqrt{6} +\sqrt{5}} = \sqrt{6} -\sqrt{5} \\\ \\ \frac{1}{\sqrt{7} +\sqrt{6}}=\sqrt{7} -\sqrt{6} \\\ \\ \frac{1}{\sqrt{8} +\sqrt{7}}= \sqrt{8} -\sqrt{7} \\\ \\ \frac{1}{\sqrt{9} +\sqrt{8}}= \sqrt{9} -\sqrt{8} \\\ \\ \\\ \\ Then \ put\ values \\\ \\⟹ \sqrt{4} -\sqrt{3}+ \sqrt{5} -\sqrt{4}+ \sqrt{6} -\sqrt{5}+ \sqrt{7} -\sqrt{6}+ \sqrt{8} -\sqrt{7}+ \sqrt{9} -\sqrt{8} \\\ \\⟹ \sqrt{9} -\sqrt{3} \\\ \\ ⟹ 3- \sqrt{3}

Option (c) is the right answer

Question 04
Simplify \ \frac{1}{\sqrt{100} -\sqrt{99}} -\frac{1}{\sqrt{99} -\sqrt{98}} +\frac{1}{\sqrt{98} -\sqrt{97}} -\frac{1}{\sqrt{97} -\sqrt{96}} +........+\frac{1}{\sqrt{2} -\sqrt{1}} \\\ \\ (a) 10 \\ \\ (b) 9 \\ \\ (c) 13 \\ \\ (d) 11

\frac{1}{\sqrt{100} -\sqrt{99}} \times \frac{\sqrt{100} +\sqrt{99}}{\sqrt{100} +\sqrt{99}} \\\ \\ ⟹ \frac{\sqrt{100} +\sqrt{99}}{100+\sqrt{9900} -\sqrt{9900} -99} \\\ \\ ⟹ \frac{\sqrt{100} +\sqrt{99}}{1} \\\ \\ ⟹ \sqrt{100} +\sqrt{99} \\\ \\ \\\ \\ Similarly \\ \\ \frac{1}{\sqrt{99} -\sqrt{98}}= \sqrt{99} +\sqrt{98} \\\ \\ \frac{1}{\sqrt{98} -\sqrt{97}}= \sqrt{98} +\sqrt{97} \\\ \\ \frac{1}{\sqrt{97} -\sqrt{96}}= \sqrt{97} +\sqrt{96} \\\ \\ \frac{1}{\sqrt{2} -\sqrt{1}}= \sqrt{2} +1 \\\ \\ \\\ \\ Now, \\ \\ \left(\sqrt{100} +\sqrt{99}\right)-( \sqrt{99} +\sqrt{98})+( \sqrt{98} +\sqrt{97})-(\sqrt{97} +\sqrt{96})+.......+( \sqrt{2} +1) \\\ \\ ⟹ \sqrt{100} +\sqrt{99} - \sqrt{99} -\sqrt{98}+ \sqrt{98} +\sqrt{97} - \sqrt{97} -\sqrt{96}+......+ \sqrt{2} +1 \\\ \\ ⟹ \sqrt{100} +1\\ \\⟹ 10+1=11

Option (d) is the right answer

Question 05

\left[\frac{1}{\sqrt{2} +\sqrt{3} -\sqrt{5}} +\frac{1}{\sqrt{2} -\sqrt{3} -\sqrt{5}}\right] \\\ \\ (a) 1 \\ \\ (b) \sqrt{2} \\ \\ (c) \frac{1}{\sqrt{2}} \\ \\ (d) 0

\left[\frac{1}{\sqrt{2} +\sqrt{3} -\sqrt{5}} +\frac{1}{\sqrt{2} -\sqrt{3} -\sqrt{5}}\right] \\\ \\ ⟹\frac{1}{\left(\sqrt{2} +\sqrt{3}\right) -\sqrt{5}} \times \frac{\left(\sqrt{2} +\sqrt{3}\right) +\sqrt{5}}{\left(\sqrt{2} +\sqrt{3}\right) +\sqrt{5}} \\\ \\ ⟹ \frac{\sqrt{2} +\sqrt{3} +\sqrt{5}}{2+3+2\sqrt{6} -5} \\\ \\ ⟹ \frac{\sqrt{2} +\sqrt{3} +\sqrt{5}}{2\sqrt{6}} \\\ \\ \\\ \\ Similarly, \\ \\ \frac{1}{\sqrt{2} -\sqrt{3} -\sqrt{5}}= \frac{-\sqrt{2} -\sqrt{3} +\sqrt{5}}{-2\sqrt{6}} \\\ \\ Now,\\ \\ \frac{\sqrt{2} +\sqrt{3} +\sqrt{5}}{2\sqrt{6}}- \frac{-\sqrt{2} -\sqrt{3} +\sqrt{5}}{-2\sqrt{6}} \\\ \\ ⟹\frac{\sqrt{2} +\sqrt{3} +\sqrt{5} -\sqrt{2} +\sqrt{3} -\sqrt{5}}{2\sqrt{6}} \\\ \\ ⟹ \frac{2\sqrt{3}}{2\sqrt{6}}= \frac{1}{\sqrt{2}}

Option (c) is the right answer

Question 06
Which is the greatest
\\\ \\ \left(\sqrt{19} -\sqrt{17}\right) ,\left(\sqrt{13} -\sqrt{11}\right),\\ \\ \left(\sqrt{7} -\sqrt{5}\right) and\left(\sqrt{5} -\sqrt{3}\right)? \\\ \\ \\\ \\ (a) \sqrt{19} -\sqrt{17} \\ \\ (b) \sqrt{13} -\sqrt{11} \\ \\ (c) \sqrt{7} -\sqrt{5} \\ \\ (d) \sqrt{5} -\sqrt{3} \\ \\

\left(\sqrt{19} -\sqrt{17}\right) \\\ \\ ⟹ \left(\sqrt{19} -\sqrt{17}\right) \times \frac{\sqrt{19} +\sqrt{17}}{\sqrt{19} +\sqrt{17}} \\\ \\ ⟹ \frac{\left(\sqrt{19} -\sqrt{17}\right)\left(\sqrt{19} +\sqrt{17}\right)}{\sqrt{19} +\sqrt{17}} \\\ \\ ⟹  \frac{19-\sqrt{323} +\sqrt{323} -17}{\sqrt{19} +\sqrt{17}} \\\ \\ ⟹ \frac{2}{\sqrt{19} +\sqrt{17}} \\\ \\ \\\ \\ Similarly, \\ \\ \left(\sqrt{13} -\sqrt{11}\right)= \frac{2}{\sqrt{13} +\sqrt{11}} \\\ \\ \sqrt{7} -\sqrt{5}= \frac{2}{\sqrt{7} +\sqrt{5}} \\\ \\ \sqrt{5} -\sqrt{3}= \frac{2}{\sqrt{5} +\sqrt{3}} \\\ \\ \sqrt{5} -\sqrt{3} is \ greatest \ because \\ \\ in \ above \ same \ numerator \ is \ divided \ by \ smallest \ denominator

Option (d) is the right answer

Question 07
Find the greatest number
\sqrt{2} ,\sqrt[6]{3} ,\sqrt[3]{4} ,\sqrt[4]{5} \\\ \\ (a) \sqrt{2} \\ \\ (b) \sqrt[6]{3} \\ \\ (c) \sqrt[3]{4} \\ \\ (d) \sqrt[4]{5}

\sqrt{2} ,\sqrt[6]{3} ,\sqrt[3]{4} ,\sqrt[4]{5} \\\ \\ ⟹ 2^{\frac{1}{2}} ,3^{\frac{1}{6}} ,4^{\frac{1}{3}} ,5^{\frac{1}{4}} \\\ \\ ⟹ 2^{\frac{6}{12}} ,3^{\frac{2}{12}} ,4^{\frac{4}{12}} ,5^{\frac{3}{12}} \ \ \ \ \ [LCM of 2,6,3,4 is 12] \\\ \\ ⟹ \left( 2^{6}\right)^{\frac{1}{12}} ,\left( 3^{2}\right)^{\frac{1}{12}} ,\left( 4^{4}\right)^{\frac{1}{12}} ,\left( 5^{3}\right)^{\frac{1}{12}} \\\ \\ ⟹ ( 64)^{\frac{1}{12}} ,( 9)^{\frac{1}{12}} ,( 256)^{\frac{1}{12}} ,( 125)^{\frac{1}{12}} \\\ \\ \sqrt[3]{4} \ is \ greatest

Option (c) is the right answer

Question 08
Find the greatest number
\sqrt{4} ,\sqrt[3]{4} ,\sqrt[4]{6} \ and \ \sqrt[6]{8} \ is \\\ \\ (a) \sqrt{4} \\ \\ (b) \sqrt[3]{4} \\ \\ (c) \sqrt[4]{6} \\ \\ (d) \sqrt[6]{8}

\sqrt{4} ,\sqrt[3]{4} ,\sqrt[4]{6} \ ,\sqrt[6]{8} \\\ \\ ⟹ 4^{\frac{1}{2}} ,4^{\frac{1}{3}} ,4^{\frac{1}{4}} ,8^{\frac{1}{6}} \\\ \\ ⟹ 4^{\frac{6}{12}} ,4^{\frac{4}{12}} ,6^{\frac{3}{12}} ,8^{\frac{2}{12}} \ \ [LCM of 2,3,4,6 is 12] \\\ \\ ⟹ \left( 4^{6}\right)^{\frac{1}{12}} ,\left( 4^{4}\right)^{\frac{1}{12}} ,\left( 6^{3}\right)^{\frac{1}{12}} ,\left( 8^{2}\right)^{\frac{1}{12}} \\\ \\ ⟹ \ ( 4096)^{\frac{1}{12}} ,( 256)^{\frac{1}{12}} ,( 216)^{\frac{1}{12}} ,( 64)^{\frac{1}{12}} \\\ \\ \sqrt{4} \ is \ greatest

Option (a) is the right answer

Question 09

\frac{12}{3+\sqrt{5} +2\sqrt{2}} \ is \ equal \ to \\\ \\ (a) 1-\sqrt{5} +\sqrt{2} +\sqrt{16} \\\ \\ (b) 1+\sqrt{5} +\sqrt{2} -\sqrt{10} \\\ \\ (c) 1+\sqrt{5} +\sqrt{2} +\sqrt{10} \\\ \\ (d) 1-\sqrt{5} -\sqrt{2} +\sqrt{10} \\\ \\ Read Solution

\frac{12}{3+\sqrt{5} +2\sqrt{2}} \\\ \\ ⟹ \frac{12}{\left( 3+\sqrt{5}\right) +2\sqrt{2}} \times \frac{\left( 3+\sqrt{5}\right) -2\sqrt{2}}{\left( 3+\sqrt{5}\right) +2\sqrt{2}} \\\ \\ ⟹\frac{12\left[\left( 3+\sqrt{5}\right) -2\sqrt{2}\right]}{\left[\left( 3+\sqrt{5}\right) +2\sqrt{2}\right]\left[\left( 3+\sqrt{5}\right) -2\sqrt{2}\right]} \\\ \\ ⟹ \frac{12\left( 3+\sqrt{5} -2\sqrt{2}\right)}{\left( 3+\sqrt{5}\right)^{2} -\left( 2\sqrt{2}\right)^{2}} \\\ \\ ⟹ \frac{12\left( 3+\sqrt{5} -2\sqrt{2}\right)}{9-8+5+6\sqrt{5}} \\\ \\ ⟹ \frac{12\left( 3+\sqrt{5} -2\sqrt{2}\right)}{6\sqrt{5} +6} \\\ \\ ⟹ \frac{12\left( 3+\sqrt{5} -2\sqrt{2}\right)}{6\left(\sqrt{5} +1\right)} \\\ \\ ⟹ \frac{2\left( 3+\sqrt{5} -2\sqrt{2}\right)}{\sqrt{5} +1} \\\ \\⟹ \frac{2\left( 3+\sqrt{5} -2\sqrt{2}\right)}{\sqrt{5} +1} \times \frac{\sqrt{5} -1}{\sqrt{5} -1} \\\ \\ ⟹ \frac{2\left( 3+\sqrt{5} -2\sqrt{2}\right)\left(\sqrt{5} -1\right)}{\left(\sqrt{5} +1\right)\left(\sqrt{5} -1\right)} \\\ \\ ⟹ \frac{2\left( 3\sqrt{5} +5-2\sqrt{10} -3-\sqrt{5} +2\sqrt{2}\right)}{5-1} \\\ \\ ⟹ \frac{2\left( 2\sqrt{5} +2+2\sqrt{2} -2\sqrt{10}\right)}{4} \\\ \\ ⟹ \frac{4\left(\sqrt{5} +1+\sqrt{2} -\sqrt{10}\right)}{4} \\\ \\ ⟹ 1+\sqrt{5} +\sqrt{2} -\sqrt{10}

Option (b) is the right answer

Question 10

\sqrt{8-2\sqrt{15}} \ is \ equal \ to: \\\ \\ (a) \sqrt{5} +\sqrt{3} \\ \\ (b) 5-\sqrt{3} \\ \\ (c) \sqrt{5} -\sqrt{3} \\ \\ (d) 3-\sqrt{5} \\\ \\ Read Solution

\sqrt{8-2\sqrt{15}} \\\ \\ ⟹ \sqrt{\left(\sqrt{5}\right)^{2} +\left(\sqrt{3}\right)^{2} -2\sqrt{5}\sqrt{3}} \\\ \\ ⟹ \sqrt{\left(\sqrt{5} -\sqrt{3}\right)^{2}} \\\ \\ ⟹ \sqrt{5} -\sqrt{3}

Option (c) is the right answer